Approximation of elliptic boundary-value problems:
Gespeichert in:
| 1. Verfasser: | |
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| Format: | Buch |
| Sprache: | English |
| Veröffentlicht: |
Mineola, NY
Dover Publ.
2007
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| Ausgabe: | Unabridged republ. of the reprint ed. with corr. in 1980 |
| Schlagworte: | |
| Online-Zugang: | Beschreibung für Leser Inhaltsverzeichnis |
| Beschreibung: | XVII, 356 S. graph. Darst. |
| ISBN: | 0486457915 |
Internformat
MARC
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| 100 | 1 | |a Aubin, Jean-Pierre |e Verfasser |4 aut | |
| 245 | 1 | 0 | |a Approximation of elliptic boundary-value problems |c Jean-Pierre Aubin |
| 250 | |a Unabridged republ. of the reprint ed. with corr. in 1980 | ||
| 264 | 1 | |a Mineola, NY |b Dover Publ. |c 2007 | |
| 300 | |a XVII, 356 S. |b graph. Darst. | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
| 650 | 0 | 7 | |a Numerische Mathematik |0 (DE-588)4042805-9 |2 gnd |9 rswk-swf |
| 650 | 0 | 7 | |a Partielle Differentialgleichung |0 (DE-588)4044779-0 |2 gnd |9 rswk-swf |
| 650 | 0 | 7 | |a Elliptische Differentialgleichung |0 (DE-588)4014485-9 |2 gnd |9 rswk-swf |
| 650 | 0 | 7 | |a Approximation |0 (DE-588)4002498-2 |2 gnd |9 rswk-swf |
| 650 | 0 | 7 | |a Randwertproblem |0 (DE-588)4048395-2 |2 gnd |9 rswk-swf |
| 650 | 0 | 7 | |a Elliptisches Randwertproblem |0 (DE-588)4193399-0 |2 gnd |9 rswk-swf |
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| 689 | 0 | 2 | |a Approximation |0 (DE-588)4002498-2 |D s |
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| 689 | 1 | 0 | |a Partielle Differentialgleichung |0 (DE-588)4044779-0 |D s |
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| 883 | 1 | |8 2\p |a cgwrk |d 20201028 |q DE-101 |u https://d-nb.info/provenance/plan#cgwrk | |
| 883 | 1 | |8 3\p |a cgwrk |d 20201028 |q DE-101 |u https://d-nb.info/provenance/plan#cgwrk | |
Datensatz im Suchindex
| _version_ | 1804139000815943680 |
|---|---|
| adam_text | Contents
Introduction
1
1
Aim and Scope
1
2
Neumann Problems
2
3
Introduction of Internal Approximations
5
4
Properties of Internal Approximations
7
5
Stability, Optimal Stability, and Regularity of the Convergence
9
6
The Case of Operators Mapping a Hubert Space onto Its Dual
10
7
Finite-Element Approximations of Sobolev Spaces
12
8
Approximation of Nonhomogeneous Neumann Problems
14
9
Approximations of Nonhomogeneous Dirichlet Problems
16
10
A Posteriori Error Estimates
20
11
External and Partial Approximations
21
12
General Outline
24
1
Approximation of Solutions of Neumann Problems for Second-Order
Linear Differential Equations
25
1
Weak Solutions of Neumann Problems for Second-Order
Linear Differential Operators
25
1-1
The Neumann Boundary-Value Problem
25
1-2
Definition of Distributions
26
1-3
Weak Derivatives of a Distribution
26
1-4
Variational Formulation of the Problem
27
1-5
Weak Solutions of the Neumann Boundary-Value Problem
27
1-6
Sobolev Spaces
29
1-7
The Lax-Milgram Theorem
30
ix
X
CONTENTS
2 Approximation
of an Abstract
Variational Problem 31
2-1 The Galerkin Approximation
of a Separable Hubert
Space
31
2-2
Approximation of a Hubert Space
32
2-3
Internal Approximation of a Variational Equation
33
2-4
Existence, Uniqueness, and Convergence Properties
34
2-5
Estimates of Global Error
35
2-6
What Kind of Approximations Should Be Chosen?
36
3
Examples of Approximations of Sobolev Spaces
37
3-1
Piecewise-Linear Approximations of the Sobolev Space
H I)
37
3-2
Estimates of Error Functions of Piecewise-Linear Approx¬
imations
38
4
Examples of Approximate Equations
40
4-1
Construction of a Finite-Difference Scheme
40
4-2
A Simpler Finite-Difference Scheme
43
2
Approximations ofHilbert Spaces
45
Hubert Spaces and Their Duals
45
1-1
Dual of a Hubert Space and Canonical Isometry
46
1-2
Example: Finite-Dimensional Hubert Spaces
47
1-3
Hahn-Banach Theorem
47
1-4
Dual of a Dense Subspace
48
1-5
Imbedding of a Space into Its Dual
49
1-6
Example: Imbedding of Spaces of Functions into Spaces
of Distributions
50
1 -7
Dual of Closed Subspaces and Factor Spaces
51
1-8
Applications to Error Estimates
52
1 -9
Dual of a Product
53
1-Ю
Dual of Domains of Operators
54
1-11
Examples: Dual of Sobolev Spaces H,m(I)
55
1-12
Properties of Bounded Sets of Operators; Uniform
Boundedness
56
1-13
Banach Theorem
57
1-14
Dual of Sobolev Spaces
Η*(Γ)
58
1-15
The Riesz-Fredholm Alternative
60
1-16
K-Elliptic and Coercive Operators
60
CONTENTS
ХІ
2
Quasi-Optimal Approximations
62
2-1
Stability Functions
63
2-2
Duality Relations between Error and Stability Functions
63
2-3
Estimates of the Stability Functions
64
2-4
Quasi-Optimal Approximations; Estimate of the Error
Function
65
2-5
Truncation Errors and Error Functions
66
3
Optimal Approximations
66
3-1
Eigenvalues and Eigenvectors of Symmetric Compact
Operators
67
3-2
Optimal Galerkin Approximations
68
3-3
Convergence and Optimality Properties
69
3-4
Spaces He
70
4
Optimal Restrictions and Prolongations; Applications
72
4-1
Optimal Restrictions and Prolongations
73
4-2
Dual Approximations
74
4-3
Construction of Optimal Prolongations and Restrictions
75
4-4
Miscellaneous Remarks
76
4-5
Characterization of Error and Stability Functions
78
4-6
Spaces of Order
0 80
3
Approximation of Operators
82
1
Internal Approximations
82
1-1
Construction of an Internal Approximate Equation
83
1-2
The Case of Finite-Dimensional Discrete Spaces
84
1-3
The Case of Operators from V onto V
84
1-4
Stability of Internal Approximations of Operators
85
1-5
Convergence and Error Estimates
86
1-6
Approximation of a Sum of an Isomorphism and a
Compact Operator
88
1-7
Approximation of Coercive and K-Elliptic Operators
90
1-8
Optimal and Quasi-Optimal Stability
92
2
Regularity of the Convergence and Estimates of Error in
Terms of
л
-Width
95
2-1
Stability and Convergence in Smaller Spaces
95
2-2
Stability and Convergence in Larger Spaces
98
2-3
Approximation of the Value of a Functional
ata
Solution
101
Xli CONTENTS
3
Discrete Convergence, Consistency, and Optimal Approxi¬
mation of Linear Operators
102
3-1
Discrete Convergence and Consistency
103
3-2
Optimal Approximation of Operators and Internal
Approximations
106
3-3
Estimates of Error and Discrete Errors
107
4
Finite-Element Approximation of Functions of One Variable
109
1
Approximation of Functions of L? by Step Functions and by
Convolution
109
1-1
The Space I? and the Discrete Space Lh2
110
1-2
The Prolongations ph°
110
1-3
The Restrictions rh
110
1-4
The Theorem of Convergence
111
1-5
Convolution of Functions and Measures
112
1-6
Approximation by Convolution
115
2
Piecewise-Polynomial Approximations of Sobolev Spaces Hm
116
2-1
Finite-Difference Operators
116
2-2
Construction of Approximations of the Space if
117
2-3
Convergence Theorem
118
2-4
Explicit Form of Functions nm
119
2-5
Properties of the Prolongations
рћт
121
2-6
Estimates of the Stability Functions
123
2-7
Optimal Properties of Prolongations phm
124
3
Finite-Element Approximations of Sobolev Spaces Hm
124
3-1
Finite-Element Approximations
125
3-2
The Criterion of m-Convergence
128
3-3
Characterization of Convergent Finite-Element Approxi¬
mations
130
3-4
Stability Properties of Finite-Element Approximations
133
5
Finite-Element Approximation of Functions of Several Variables
135
1
Approximations of the Sobolev Spaces
Ят(/гв)
136
1-1
Notations
136
1-2
Finite-Element Approximations
138
1-3
(2w
+
l^-Level Piecewise-Polynomial Approximations
141
1-4
[2(2m)n -(2m- l)n]-Level Piecewise-Polynomial
Approximations
142
CONTENTS
ХІІІ
2
Approximations
of the Sobolev Spaces
Нт(Џ)
149
2-1
Sobolev Spaces
Нт(п)
149
2-2
Finite-Element Approximations of Hm(Q)
150
2-3
Quasi-Optimal Finite-Element Approximations of
/Ρ^Ω)
151
2-4
Piecewise-Polynomial Approximations of
Η™(Ώ)
154
3
Approximation of the Sobolev Spaces
Нот(П)
157
3-1
Sobolev Spaces
#om(Ü) 157
3-2
Finite-Element Approximations of Hom(Ci)
159
3-3
Convergent Finite-Element Approximations of Hom(fi)
159
6
Boundary-Value Problems and the Trace Theorem
162
1
Some Variational Boundary-Value Problems for the Laplacian
162
1-1
The Laplacian
163
1-2
Characterization of Sobolev Spaces H^Q)
163
1-3
The Green Formula
f
164
1-4
The Dirichlet Problem for the Laplacian
165
1-5
The Neumann Problem for the Laplacian
166
1-6
A Mixed Problem for the Laplacian
167
1-7
An Oblique Problem for the Laplacian
168
1-8
Existence and Uniqueness of the Solutions
170
2
Variational Boundary-Value Problems and Their
Adjoints
170
2-1
Spaces
К, Я
and Operator
y
171
2-2
Formal Operator
Л
Associated with a(u, v)
172
2-3
The Green Formula
172
2-4
Abstract Neumann and Dirichlet Problems Associated
with a(u, v)
174
2-5
Mixed Type Boundary-Value Problems Associated with
a(u, v)
175
2-6
Existence and Uniqueness of the Solutions of Boundary-
Value Problems
178
2-7
Formal Adjoint of an Operator and Green s Formula
182
2-8
Theorems of Regularity
184
3
The Trace Theorem and Properties of Sobolev Spaces
187
3-1
Statement of the Trace Theorem
187
3-2
Change of Coordinates
188
3-3
Sobolev Spaces
Я (А )
for Real Numbers
s
189
3-4
Sobolev Spaces
Я«(Г)
and
Η%Ω)
190
3-5
Trace Operators and Operators of Extension
:
Theorems
of Density
191
xiv
CONTENTS
3-6
Properties of the Spaces If^R )
194
3-7
Proof of the Trace Theorem
196
3-8
Sobolev Inequalities and the Trace Theorem in Space
ЩЇЇ)
198
3-9
Theorem of Compactness
199
7
Examples of Boundary-Value Problems
201
1
Boundary-Value Problems for Second-Order Differential
Operators
201
1-1
Second-Order Linear Differential Operators
201
1-2
Elliptic Second-Order Partial Differential Operators
202
1-3
The Dirichlet Problem
203
1-4
The Neumann Problem
204
1-5
Mixed Problems
204
1-6
Oblique Problems
205
1-7
Interface Problems
206
1-8
The Regularity Theorem
208
1 -9
Theorems of Isomorphism
211
1-10
Value of the Solution at a Point of the Boundary
212
1-11
Problems with Elliptic Differential Boundary Conditions
213
2
Boundary-Value Problems for Differential Operators of
Higher Order
214
2-1
Linear Differential Operators of Order 2k
214
2-2
The Dirichlet Problem
215
2-3
The Neumann Problem
215
2-4
Regularity and Theorems of Isomorphism
216
2-5
Other Boundary-Value Problems
217
2-6
Boundary Value Problems for
Δ2
+
λ
218
8
Approximation of Neumann-Type Problems
222
1
Theorems of Convergence and Error Estimates
222
1-1
Internal Approximation of a Neumann-type Problem
223
1 -2
Convergence and Estimates of Error in Larger Spaces
225
2
Approximation of Neumann Problems for Elliptic Operators
of Order 2k
229
2-1
Approximation of Neumann Problems for Elliptic
Differential Operators
233
2-2
Convergence Properties of Finite Element Approxi¬
mations of Neumann Problems
233
CONTENTS
XV
2-3 The
(2и!
+
l^-Level Approximations
of the Neumann
Problem
235
2-4
The [2(2re)n
-
(2m
—
l)n]-Level Approximations of the
Neumann Problem
238
2-5
Approximations of the Spaces
Η (Ω, Λ)
and
#(Ω, Λ)
239
3
Approximation of Other Neumann-Type Problems
240
3-1
Approximation of the Value of the Solution at a Point
of the Boundary
240
3-2
Approximation of Oblique Boundary-Value Problems
242
3-3
Approximation of a Problem with Elliptic Boundary
Conditions
243
3-4
Approximation of Interface Problems
248
3-5
Approximation of the Neumann Problem for
Δ2 + γ
250
9
Perturbed Approximations and Least-Squares Approximations
252
1
Perturbed Approximations
252
1-1
Internal Approximation of a Variational Boundary-
Value Problem
253
1-2
Perturbed Approximation of a Variational Boundary-
Value Problem
254
1-3
Convergence in the Initial Space
255
1-4
Estimates of Error
257
1-5
Convergence in Smaller Spaces
259
1-6
Convergence in Larger Spaces
259
2
Perturbed Approximations of Boundary-Value Problems
261
2-1
Perturbed Approximations by Finite-Element Approxi¬
mations
261
2-2
Error Estimates and Regularity of the Convergence
264
2-3
The 3 -level Perturbed Approximation of the Dirichlet
Problem
265
3
Least-Squares Approximations
267
3-1
Least-Squares Approximation Schemes
267
3-2
Error Estimates (I)
269
3-3
Error Estimates (II)
270
3-4
Least-Squares Approximations of Dirichlet Problems
275
CONTENTS XVI
10
Conjugate Problems and A Posteriori Error Estimates
280
1
Conjugate Problems of Boundary-Value Problems
280
1-1
First Example of a Conjugate Problem
280
1-2
Second Example of a Conjugate Problem
282
1-3
Construction of Conjugate Problems
285
2
Applications to the Approximation of Dirichlet Problems
293
2-1
Approximation of the Dirichlet Problem (I)
293
2-2
Approximation of the Dirichlet Problem (II)
295
2-3
The Case of Second-Order Differential Operators
296
3
Finite-Element Approximations of the Spaces
Η*(Ω,
D*)
297
3-1
Spaces
Η*(Ω,
D*)
297
3-2
Approximations of the Space
Η*(Ω,
D*)
298
4
Approximation of the Second Example of a Conjugate
Problem
302
4-1
Approximation of the Conjugate Dirichlet Problem
302
4-2
Properties of the Discrete Conjugate Problem
306
11
External and Partial Approximations
307
1
External Approximations; Stability, Convergence, and Error
Estimates
307
1-1
Definition of External Approximations
307
1-2
Example: Partial Approximations of a Finitelntersection
of Spaces
310
1-3
Stability and Convergence of External Approximations
of Operators
311
1-4
Estimates of Error and Regularity of the Convergence
312
1-5
Properties of the External Error Functions
314
2
External and Partial Approximations of Variational Equations
317
2-1
Partial Approximation of a Split Variational Equation
317
2-2
External Approximation of Variational Equations
319
2-3
Partial Approximation of Neumann Problems
321
2-4
Perturbed Partial Approximation of Boundary-Value
Problems
324
CONTENTS
XVII
3
Partial Approximations of Sobolev Spaces
328
3-1
Spaces
#(Ω,
Dt)
328
3-2
Partial Approximations of the Sobolev Space
Н Сї)
329
3-3
Estimates of Truncation Errors and External Error
Functions
332
3-4
Partial Approximations of the Sobolev Spaces
Hm(ß)
and
Яот(П)
334
4
Partial Approximation of Boundary-Value Problems
336
4-1
Partial Approximation of Second-Order Linear Operators
336
4-2
Partial Approximation of the Neumann Problem
338
4-3
Perturbed Partial Approximation of Mixed Boundary-
Value Problems
340
4-4
Estimates of Error in the Interior
342
4-5
Partial Approximations of Higher-Order Differential
Operators
343
Comments
346
References
349
Index
355
|
| any_adam_object | 1 |
| author | Aubin, Jean-Pierre |
| author_facet | Aubin, Jean-Pierre |
| author_role | aut |
| author_sort | Aubin, Jean-Pierre |
| author_variant | j p a jpa |
| building | Verbundindex |
| bvnumber | BV023802372 |
| classification_rvk | SK 540 |
| ctrlnum | (OCoLC)636595852 (DE-599)BVBBV023802372 |
| discipline | Mathematik |
| edition | Unabridged republ. of the reprint ed. with corr. in 1980 |
| format | Book |
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| id | DE-604.BV023802372 |
| illustrated | Illustrated |
| indexdate | 2024-07-09T21:37:08Z |
| institution | BVB |
| isbn | 0486457915 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-017444571 |
| oclc_num | 636595852 |
| open_access_boolean | |
| owner | DE-634 DE-703 DE-83 |
| owner_facet | DE-634 DE-703 DE-83 |
| physical | XVII, 356 S. graph. Darst. |
| publishDate | 2007 |
| publishDateSearch | 2007 |
| publishDateSort | 2007 |
| publisher | Dover Publ. |
| record_format | marc |
| spelling | Aubin, Jean-Pierre Verfasser aut Approximation of elliptic boundary-value problems Jean-Pierre Aubin Unabridged republ. of the reprint ed. with corr. in 1980 Mineola, NY Dover Publ. 2007 XVII, 356 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier Numerische Mathematik (DE-588)4042805-9 gnd rswk-swf Partielle Differentialgleichung (DE-588)4044779-0 gnd rswk-swf Elliptische Differentialgleichung (DE-588)4014485-9 gnd rswk-swf Approximation (DE-588)4002498-2 gnd rswk-swf Randwertproblem (DE-588)4048395-2 gnd rswk-swf Elliptisches Randwertproblem (DE-588)4193399-0 gnd rswk-swf Elliptische Differentialgleichung (DE-588)4014485-9 s Randwertproblem (DE-588)4048395-2 s Approximation (DE-588)4002498-2 s 1\p DE-604 Partielle Differentialgleichung (DE-588)4044779-0 s Numerische Mathematik (DE-588)4042805-9 s 2\p DE-604 Elliptisches Randwertproblem (DE-588)4193399-0 s 3\p DE-604 http://www.loc.gov/catdir/enhancements/fy0702/2006052124-d.html Beschreibung für Leser Digitalisierung UB Bayreuth application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=017444571&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis 1\p cgwrk 20201028 DE-101 https://d-nb.info/provenance/plan#cgwrk 2\p cgwrk 20201028 DE-101 https://d-nb.info/provenance/plan#cgwrk 3\p cgwrk 20201028 DE-101 https://d-nb.info/provenance/plan#cgwrk |
| spellingShingle | Aubin, Jean-Pierre Approximation of elliptic boundary-value problems Numerische Mathematik (DE-588)4042805-9 gnd Partielle Differentialgleichung (DE-588)4044779-0 gnd Elliptische Differentialgleichung (DE-588)4014485-9 gnd Approximation (DE-588)4002498-2 gnd Randwertproblem (DE-588)4048395-2 gnd Elliptisches Randwertproblem (DE-588)4193399-0 gnd |
| subject_GND | (DE-588)4042805-9 (DE-588)4044779-0 (DE-588)4014485-9 (DE-588)4002498-2 (DE-588)4048395-2 (DE-588)4193399-0 |
| title | Approximation of elliptic boundary-value problems |
| title_auth | Approximation of elliptic boundary-value problems |
| title_exact_search | Approximation of elliptic boundary-value problems |
| title_full | Approximation of elliptic boundary-value problems Jean-Pierre Aubin |
| title_fullStr | Approximation of elliptic boundary-value problems Jean-Pierre Aubin |
| title_full_unstemmed | Approximation of elliptic boundary-value problems Jean-Pierre Aubin |
| title_short | Approximation of elliptic boundary-value problems |
| title_sort | approximation of elliptic boundary value problems |
| topic | Numerische Mathematik (DE-588)4042805-9 gnd Partielle Differentialgleichung (DE-588)4044779-0 gnd Elliptische Differentialgleichung (DE-588)4014485-9 gnd Approximation (DE-588)4002498-2 gnd Randwertproblem (DE-588)4048395-2 gnd Elliptisches Randwertproblem (DE-588)4193399-0 gnd |
| topic_facet | Numerische Mathematik Partielle Differentialgleichung Elliptische Differentialgleichung Approximation Randwertproblem Elliptisches Randwertproblem |
| url | http://www.loc.gov/catdir/enhancements/fy0702/2006052124-d.html http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=017444571&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| work_keys_str_mv | AT aubinjeanpierre approximationofellipticboundaryvalueproblems |